Résumé:
This thesis deals with the problem of scalar particle creation in time-dependent systems using
the Ermakov-Lewis invariant method. We begin by reviewing di erent formulations of quantum
mechanics, with an emphasis on those that can be extended to quantum eld theory. We then study
particle creation in time-dependent electric elds, making use of solutions of the Ermakov{Milne
equation, and discuss several exactly solvable cases.
We next turn to the cosmological setting and consider the dynamics of quantum elds in an
expanding universe, both isotropic and anisotropic. In this context, we show that the time-dependent
particle number can be interpreted as a measure of the deviation between the adiabatic vacuum
and the invariant vacuum. We also point out that a suitable rede nition of the elds, together with
a reparameterization of time, is often necessary to recover an adiabatic evolution in the asymptotic
regimes, and hence to give a consistent de nition of particles.
Finally, we introduce a uniform semiclassical approximation to deal with more complicated
situations where exact analytical solutions are not available, such as transitions between di erent
cosmological eras or models involving a generalized uncertainty principle with a minimal length.
This work shows that the Ermakov-Lewis approach provides a general and useful framework for
describing non-stationary quantum systems, and o ers a coherent way to analyze particle creation
in various physical contexts.
